A game-theoretic solution to the inconsistency between Thrasymachus and Glaucon in Plato's republic

In Book I of Plato's Republic, Thrasymachus contends two major claims: (i) justice is the adaantage of the stronger, and (ii) justice is the good of the other, while injustice is to one's own profit and adaantage. In the beginning of Book II, Glaucon self-proclaims that he will be representing Thrasymachus' claims in a better way, and proaides a story of how justice has originated from a state-of-nature situation. Howeaer, Glaucon's story of the origin of justice has an implication that justice is the adaantage of the weak rather than the stronger. This is inconsistent with Thrasymachus' first claim, which states that justice is the adaantage of the stronger. This is a problem for Glaucon since he is supposed to be representing Thrasymachus' original claims in a better way. In the present article, I proaide two solutions to this puzzle with the help of elementary game theory.